Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble

Abstract

The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble, and furthermore relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a $\beta$-generalisation of these results, where now the role of the Schur polynomials is played by the Jack polynomials. This leads to an explicit evaluation of the scaled moments for all $\beta >0$, subject to the constraint that a particular parameter therein is equal to a non negative integer. Consideration is also given to the calculation of the moments of the singular statistic ${\sum }_{j=1}^{N}1/x_j$ for the Jacobi $\beta$-ensemble.

arXiv comment29 pages; v2 guided by referee reports

References in corpus (18)

• GENERALIZED CIRCULAR ENSEMBLE OF SCATTERING MATRICES FOR A CHAOTIC CAVITY WITH NON-IDEAL LEADS
• Infinite random matrices and ergodic measures
• Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble
• Finite N Fluctuation Formulas for Random Matrices
• Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes
• Matrix models for circular ensembles
• Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages
• Recursion scheme for the largest $\beta$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport
• Derivative moments for characteristic polynomials from the CUE
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• Mixed moments of characteristic polynomials of random unitary matrices
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials
• Infinite Random Matrices and Ergodic decomposition of Finite or Infinite Hua-Pickrell measures
• Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$
• Differential recurrences for the distribution of the trace of the $\beta$-Jacobi ensemble